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Optimal sequential tests for two simple hypotheses. (English) Zbl 1162.62080
Summary: A general problem of testing two simple hypotheses about the distribution of a discrete-time stochastic process is considered. The main goal is to minimize an average sample number over all sequential tests whose error probabilities do not exceed some prescribed levels. As a criterion of minimization, the average sample number under a third hypothesis is used [modified Kiefer-Weiss problem; J. Kiefer and L. Weiss, Ann. Math. Stat. 28, 57–74 (1957; Zbl 0079.35406)]. For a class of sequential testing problems, the structure of optimal sequential tests is characterized. An application to the Kiefer-Weiss problem for discrete-time stochastic processes is proposed. As another application, the structure of Bayes sequential tests for two composite hypotheses, with a fixed cost per observation, is given. The results are also applied for finding optimal sequential tests for discrete-time Markov processes. In a particular case of testing two simple hypotheses about a location parameter of an autoregressive process of order 1, it is shown that the sequential probability ratio test has the A. Wald and J. Wolfowitz optimality property [Ann. Math. Stat. 19, 326–339 (1948; Zbl 0032.17302)].
MSC:
62L10Sequential statistical analysis
62L15Optimal stopping (statistics)
62C10Bayesian problems; characterization of Bayes procedures
62M07Non-Markovian processes: hypothesis testing
62F15Bayesian inference
62M02Markov processes: hypothesis testing